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August  2014, 19(6): 1769-1781. doi: 10.3934/dcdsb.2014.19.1769

## On the minimal speed of front propagation in a model of the Belousov-Zhabotinsky reaction

 1 Department of Differential Equations, National Technical University, Kyiv 2 Facultad de Ciencias, Universidad de Chile, Santiago, Chile 3 Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca

Received  June 2013 Revised  April 2014 Published  June 2014

In this paper, we answer the question about the existence of the minimal speed of front propagation in a delayed version of the Murray model of the Belousov-Zhabotinsky (BZ) chemical reaction. It is assumed that the key parameter $r$ of this model satisfies $0< r \leq 1$ that makes it formally monostable. By proving that the set of all admissible speeds of propagation has the form $[c_*,+\infty)$, we show here that the BZ system with $r \in (0,1]$ is actually of the monostable type (in general, $c_*$ is not linearly determined). We also establish the monotonicity of wavefronts and present the principal terms of their asymptotic expansions at infinity (in the critical case $r=1$ inclusive).
Citation: Elena Trofimchuk, Manuel Pinto, Sergei Trofimchuk. On the minimal speed of front propagation in a model of the Belousov-Zhabotinsky reaction. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1769-1781. doi: 10.3934/dcdsb.2014.19.1769
##### References:
 [1] A. Boumenir and V. M. Nguyen, Perron theorem in the monotone iteration method for traveling waves in delayed reaction-diffusion equations,, J. Differential Equations, 244 (2008), 1551. doi: 10.1016/j.jde.2008.01.004. [2] M. S. P. Eastham, The Asymptotic Solution of Linear Differential Systems,, Clarendon Press, (1989). [3] I. R. Epstein and Y. Luo, Differential delay equations in chemical kinetics. Nonlinear models: the cross-shaped phase diagram and the Oregonator,, J. Chem. Phys., 95 (1991), 244. doi: 10.1063/1.461481. [4] J. Fang and J. Wu, Monotone travelling waves for delayed Lotka-Volterra competition systems,, Discrete Contin. Dyn. Syst., 32 (2012), 3043. doi: 10.3934/dcds.2012.32.3043. [5] R. J. Field and R. M. Noyes, Oscillations in chemical systems. V. Quantitative explanation of band migration in the Belousov-Zhabotinskii reaction,, J. Am. Chem. Soc., 96 (1974), 2001. doi: 10.1021/ja00814a003. [6] S.-C. Fu, Travelling waves of a reaction-diffusion model for the acidic nitrate-ferroin reaction,, Discrete Contin. Dyn. Syst. B, 16 (2011), 189. doi: 10.3934/dcdsb.2011.16.189. [7] J.-S. Guo and X. Liang, The minimal speed of traveling fronts for the Lotka-Volterra competition system,, J. Dynam. Differential Equations, 23 (2011), 353. doi: 10.1007/s10884-011-9214-5. [8] J. Huang and X. Zou, Existence of traveling wavefronts of delayed reaction diffusion systems without monotonicity,, Discrete Contin. Dyn. Syst., 9 (2003), 925. doi: 10.3934/dcds.2003.9.925. [9] W. Huang, Problem on minimum wave speed for a Lotka-Volterra reaction-diffusion competition model,, J. Dynam. Differential Equations, 22 (2010), 285. doi: 10.1007/s10884-010-9159-0. [10] W. Huang and M. Han, Non-linear determinacy of minimum wave speed for a Lotka-Volterra competition model,, J. Differential Equations, 251 (2011), 1549. doi: 10.1016/j.jde.2011.05.012. [11] Ya. I. Kanel, The existence of a solution of traveling wave type for the Belousov-Zhabotinskii system of equations. II,, Siberian Math. J., 32 (1991), 390. [12] B. Li and L. Zhang, Travelling wave solutions in delayed cooperative systems,, Nonlinearity, 24 (2011), 1759. doi: 10.1088/0951-7715/24/6/004. [13] X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Functional Anal., 259 (2010), 857. doi: 10.1016/j.jfa.2010.04.018. [14] G. Lin and W.-T. Li, Travelling wavefronts of Belousov-Zhabotinskii system with diffusion and delay,, Appl. Math. Letters, 22 (2009), 341. doi: 10.1016/j.aml.2008.04.006. [15] G. Lin and S. Ruan, Traveling Wave Solutions for Delayed Reaction-Diffusion Systems and Applications to Lotka-Volterra Competition-Diffusion Models with Distributed Delays,, J. Dynam. Differential Equations, (2014). doi: 10.1007/s10884-014-9355-4. [16] G. Lv and M. Wang, Traveling wave front in diffusive and competitive Lotka-Volterra systems,, Nonlinear Anal. Real World Appl., 11 (2010), 1323. doi: 10.1016/j.nonrwa.2009.02.020. [17] S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem,, J. Differential Equations, 171 (2001), 294. doi: 10.1006/jdeq.2000.3846. [18] J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type,, J. Dynam. Differential Equations, 11 (1999), 1. doi: 10.1023/A:1021889401235. [19] J. D. Murray, On traveling wave solutions in a model for Belousov-Zhabotinskii reaction,, J. Theor. Biol., 56 (1976), 329. doi: 10.1016/S0022-5193(76)80078-1. [20] J. D. Murray, Lectures on Nonlinear Differential Equations, Models in biology,, Clarendon Press, (1977). [21] M. R. Roussel, The use of delay differential equations in chemical kinetics,, J. Phys. Chem., 100 (1996), 8323. doi: 10.1021/jp9600672. [22] E. Trofimchuk, M. Pinto and S. Trofimchuk, Pushed traveling fronts in monostable equations with monotone delayed reaction,, Discrete Contin. Dyn. Syst., 33 (2013), 2169. doi: 10.3934/dcds.2013.33.2169. [23] E. Trofimchuk, M. Pinto and S. Trofimchuk, Traveling waves for a model of the Belousov-Zhabotinsky reaction,, J. Differential Equations, 254 (2013), 3690. doi: 10.1016/j.jde.2013.02.005. [24] A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems,, Amer. Math. Soc., (1994). [25] J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay,, J. Dynam. Differential Equations, 13 (2001), 651. doi: 10.1023/A:1016690424892. [26] Q. Ye and M. Wang, Traveling wave front solutions of Noyes-Field System for Belousov-Zhabotinskii reaction,, Nonlinear Anal., 11 (1987), 1289. doi: 10.1016/0362-546X(87)90046-0.

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##### References:
 [1] A. Boumenir and V. M. Nguyen, Perron theorem in the monotone iteration method for traveling waves in delayed reaction-diffusion equations,, J. Differential Equations, 244 (2008), 1551. doi: 10.1016/j.jde.2008.01.004. [2] M. S. P. Eastham, The Asymptotic Solution of Linear Differential Systems,, Clarendon Press, (1989). [3] I. R. Epstein and Y. Luo, Differential delay equations in chemical kinetics. Nonlinear models: the cross-shaped phase diagram and the Oregonator,, J. Chem. Phys., 95 (1991), 244. doi: 10.1063/1.461481. [4] J. Fang and J. Wu, Monotone travelling waves for delayed Lotka-Volterra competition systems,, Discrete Contin. Dyn. Syst., 32 (2012), 3043. doi: 10.3934/dcds.2012.32.3043. [5] R. J. Field and R. M. Noyes, Oscillations in chemical systems. V. Quantitative explanation of band migration in the Belousov-Zhabotinskii reaction,, J. Am. Chem. Soc., 96 (1974), 2001. doi: 10.1021/ja00814a003. [6] S.-C. Fu, Travelling waves of a reaction-diffusion model for the acidic nitrate-ferroin reaction,, Discrete Contin. Dyn. Syst. B, 16 (2011), 189. doi: 10.3934/dcdsb.2011.16.189. [7] J.-S. Guo and X. Liang, The minimal speed of traveling fronts for the Lotka-Volterra competition system,, J. Dynam. Differential Equations, 23 (2011), 353. doi: 10.1007/s10884-011-9214-5. [8] J. Huang and X. Zou, Existence of traveling wavefronts of delayed reaction diffusion systems without monotonicity,, Discrete Contin. Dyn. Syst., 9 (2003), 925. doi: 10.3934/dcds.2003.9.925. [9] W. Huang, Problem on minimum wave speed for a Lotka-Volterra reaction-diffusion competition model,, J. Dynam. Differential Equations, 22 (2010), 285. doi: 10.1007/s10884-010-9159-0. [10] W. Huang and M. Han, Non-linear determinacy of minimum wave speed for a Lotka-Volterra competition model,, J. Differential Equations, 251 (2011), 1549. doi: 10.1016/j.jde.2011.05.012. [11] Ya. I. Kanel, The existence of a solution of traveling wave type for the Belousov-Zhabotinskii system of equations. II,, Siberian Math. J., 32 (1991), 390. [12] B. Li and L. Zhang, Travelling wave solutions in delayed cooperative systems,, Nonlinearity, 24 (2011), 1759. doi: 10.1088/0951-7715/24/6/004. [13] X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Functional Anal., 259 (2010), 857. doi: 10.1016/j.jfa.2010.04.018. [14] G. Lin and W.-T. Li, Travelling wavefronts of Belousov-Zhabotinskii system with diffusion and delay,, Appl. Math. Letters, 22 (2009), 341. doi: 10.1016/j.aml.2008.04.006. [15] G. Lin and S. Ruan, Traveling Wave Solutions for Delayed Reaction-Diffusion Systems and Applications to Lotka-Volterra Competition-Diffusion Models with Distributed Delays,, J. Dynam. Differential Equations, (2014). doi: 10.1007/s10884-014-9355-4. [16] G. Lv and M. Wang, Traveling wave front in diffusive and competitive Lotka-Volterra systems,, Nonlinear Anal. Real World Appl., 11 (2010), 1323. doi: 10.1016/j.nonrwa.2009.02.020. [17] S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem,, J. Differential Equations, 171 (2001), 294. doi: 10.1006/jdeq.2000.3846. [18] J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type,, J. Dynam. Differential Equations, 11 (1999), 1. doi: 10.1023/A:1021889401235. [19] J. D. Murray, On traveling wave solutions in a model for Belousov-Zhabotinskii reaction,, J. Theor. Biol., 56 (1976), 329. doi: 10.1016/S0022-5193(76)80078-1. [20] J. D. Murray, Lectures on Nonlinear Differential Equations, Models in biology,, Clarendon Press, (1977). [21] M. R. Roussel, The use of delay differential equations in chemical kinetics,, J. Phys. Chem., 100 (1996), 8323. doi: 10.1021/jp9600672. [22] E. Trofimchuk, M. Pinto and S. Trofimchuk, Pushed traveling fronts in monostable equations with monotone delayed reaction,, Discrete Contin. Dyn. Syst., 33 (2013), 2169. doi: 10.3934/dcds.2013.33.2169. [23] E. Trofimchuk, M. Pinto and S. Trofimchuk, Traveling waves for a model of the Belousov-Zhabotinsky reaction,, J. Differential Equations, 254 (2013), 3690. doi: 10.1016/j.jde.2013.02.005. [24] A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems,, Amer. Math. Soc., (1994). [25] J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay,, J. Dynam. Differential Equations, 13 (2001), 651. doi: 10.1023/A:1016690424892. [26] Q. Ye and M. Wang, Traveling wave front solutions of Noyes-Field System for Belousov-Zhabotinskii reaction,, Nonlinear Anal., 11 (1987), 1289. doi: 10.1016/0362-546X(87)90046-0.
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